The present invention relates to a method for efficiently computing the Fourier transform of an irregularly-sampled data set in forming multi-dimensional synthetic aperture images.
Synthetic aperture radar (SAR) may be an attractive imaging technique because of its ability to operate under all lighting conditions and through clouds and haze. FIG. 1 illustrates an example of a collection of 2-D SAR data. An airplane 1 flies past an area of interest 2 while collecting radar data. The flight path is usually in a straight line. The flight direction is called the azimuth. The direction normal from the flight path to the region of interest 2 is called the range. The plane which is formed by the azimuth and range directions is the slant plane. The normal to the slant plane is the elevation. Processing algorithms form a high-resolution 2-D image of the region of interest 2 by combining the information from all of the radar data. In doing so, the processing algorithms effectively synthesize an aperture that is much larger than the actual aperture of the antenna. Hence the name, xe2x80x9csynthetic aperture radarxe2x80x9d.
While successful in many applications, the 2-D form of SAR yields very limited information about the distribution of objects in the elevation dimension. Further, the 2-D form of SAR has limited utility in detecting and identifying objects obscured by overlying layers. FIG. 1 illustrates an example of 2-D SAR imaging of a 3-D scene that contains objects concealed by overlying foliage. The radar illuminates the scene from the left at a single elevation. The flight path is perpendicular to the plane of the page. Because a conventional SAR image is purely 2-D, the energy within a given (range, azimuth) pixel is the sum of the energy returned by all scatterers at that range and azimuth, regardless of their elevation. In three dimensions, the frequency space is a plane (as shown, for example, in FIG. 3) and the image pixels have a columnar shape (as shown, for example, in FIG. 4). Energy returned from the overlying layers (foliage, in this case) may be integrated with the energy returned from the objects below, which reduces the signal-to-clutter ratio of the objects. Resolution in the third dimension may be required to separate the desired signal from the clutter.
Three-dimensional SAR extends the synthetic aperture concept used in one dimension (azimuth) in conventional SAR to two dimensions (azimuth and elevation). An example of 3-D SAR imaging of a 3-D scene is illustrated, for example, in FIG. 5. The radar now illuminates the scene from the left at multiple elevations, which creates a synthetic aperture that has two dimensions instead of one. The frequency space from this type of collection contains multiple planes, as shown, for example, in FIG. 6. The resulting impulse response shows resolution in all three dimensions, as shown, for example, in FIG. 7. The returns from the overlying layers and the objects on the ground are contained in different voxels. Thus, the signal-to-clutter ratio is improved enabling easier detection and identification of the objects. The 2-D aperture also effectively increases the coherent integration time, which improves the signal-to-noise ratio. It is noted that interferometric SAR (IFSAR), which collects data at two elevations and is sometimes referred to as 3-D SAR, is in fact a degenerate case of true 3-D SAR.
Forming 3-D synthetic aperture (SAR) images may be a computationally-intensive task because of the irregular sampling of the data and the curvature of the sensing wavefronts. Algorithms that have been proposed for forming 3-D SAR images all have drawbacks with respect to their computational complexity or compensation for the irregular sampling or wavefront curvature. Time domain backprojection (TDBP) is the standard image formation technique because it perfectly compensates for non-ideal antenna motion and for wavefront curvature. However, the computational complexity can be extremely high. Various fast TDBP methods have been proposed that reduce the computational complexity with varying tradeoffs in image quality. The polar format algorithm (PFA) and the range migration algorithm (RMA) are much faster, but the PFA cannot compensate for wavefront curvature and the RMA cannot compensate for irregular antenna motion.
Accordingly, a need exists for an image formation technique which can compensate for irregular sampling and wavefront curvature, but which is also faster than the TDBP methods.
One embodiment of the invention includes a method of forming an N-dimensional volume, where N is not less than two, from a collected data set which comprises a plurality of single-valued subsets, comprising: forming at least one (N-1)-dimensional images from each of the single-valued subsets, respectively; performing a convolution operation on each (N-1)-dimensional image and an impulse response at different values of xN, where xN is a dimension not formed in the (N-1) dimensional image, to form respective N-dimensional subset volumes; and combining the N-dimensional subset volumes to form the N-dimensional volume.
In one aspect of the invention, the N-dimensional volume is formed at additional values of xN.
In another aspect of the invention, kxe2x80x21:N-1 is calculated in an analytical form and the analytical form of kxe2x80x21:N-1 is evaluated on a regular rectilinear grid to determine the impulse response for each value of xN.
Another aspect of the invention comprises calculating kxe2x80x21:N-1 on an irregular grid; and interpolating kxe2x80x21:N-1 onto a regular rectilinear grid to determine the impulse response for each value of xN.
A further aspect of the invention comprises calculating a frequency space phase xcfx86 of the impulse response at one or more values of xN to determine the impulse response for each value of xN.
Another aspect of the invention comprises calculating the frequency space phase xcfx86 of the impulse response by forming the impulse response at one or more values of xN; calculating a Fourier transform of the impulse response at each value of xN; and calculating a phase of each Fourier transform.
Another aspect of the invention comprises calculating the frequency space phase xcfx86 of the impulse response by interpolating irregularly-sampled impulse frequency space data at one or more values of xN onto a regularly-sampled grid.
A further aspect of the invention comprises forming the impulse response at two or more values of xN; calculating a Fourier transform for each impulse response; dividing one of the Fourier transforms by another one of the Fourier transforms; and calculating a phase of the divisor to calculate the frequency space phase xcfx86 of the impulse response.
Another aspect of the invention comprises adjusting, as xN varies, a slope formed by kxe2x80x21:N-1 of the impulse response to compensate for an effect on the collected data set due to curved wavefronts.
Another aspect of the invention comprises calculating the impulse response from an irregularly-sampled data at one value of xN.
Another aspect of the invention comprises calculating the impulse response from an irregularly-sampled data at regular xN intervals.
Another aspect of the invention comprises determining the regular xN intervals by xN=nxcex94xN, where n is an integer.
Another aspect of the invention comprises determining the regular xN intervals by xN=xcex94xNn, where n is an integer.
Another aspect of the invention comprises calculating the impulse response from the irregularly-sampled data at irregular xN intervals.
Another aspect of the invention comprises calculating the (N-1)-dimensional images on a regular rectilinear sampling grid at one or more xN values.
Another aspect of the invention comprises calculating the (N-1)-dimensional image at one xN value.
Another aspect of the invention comprises calculating the (N-1)-dimensional image at regular xN intervals.
Another aspect of the invention comprises determining the regular xN intervals by xN=nxcex94xN, where n is an integer.
Another aspect of the invention comprises determining the regular xN intervals by xN=xcex94xNn, where n is an integer.
Another aspect of the invention comprises calculating the (N-1)-dimensional image at irregular xN intervals.
Another aspect of the invention comprises performing the convolution operation by multiplication in a frequency space.
Another aspect of the invention comprises performing the convolution operation directly in a spatial domain.
Another aspect of the invention comprises performing the convolution and combining operations as a discrete Fourier transform in the N-th dimension and as a fast Fourier transform in the other N-1 dimensions.
Another aspect of the invention comprises performing the discrete Fourier transform in the N-th dimension as an interpolation and a fast Fourier transform.
In another embodiment of the invention, a computer readable medium stores a program to form an N-dimensional volume, where N is not less than two, from a collected data set, which is divided into a plurality of single-valued subsets, comprising: forming at least one (N-1)-dimensional images from each of the single-valued subsets, respectively; performing a convolution operation on each (N-1)-dimensional image and an impulse response at different values of xN, where xN is a dimension not formed in the (N-1) dimensional image, to form respective N-dimensional subset volumes; and combining the N-dimensional subset volumes to form the N-dimensional volume.
In a further embodiment of the invention, a system comprises: a sensor to collect a data set relating to a structure of a scene; a transfer unit in communication with the sensor to receive the collected data set from the sensor; a computer connected to the transfer unit which operates a program stored thereon to form an N-dimensional volume, where N is not less than two, from the transferred collected data set, which is divided into a plurality of single-valued subsets, the program performing a process comprising forming at least one (N-1)-dimensional images from each of the single-valued subsets, respectively, performing a convolution operation on each (N-1)-dimensional image and an impulse response at different values of xN, where xN is a dimension not formed in the (N-1) dimensional image, to form respective N-dimensional subset volumes, and combining the N-dimensional subset volumes to form the N-dimensional volume; and a display to display the N-dimensional volume.
In an aspect of the invention, the sensor uses electromagnetic radiation.
In another aspect of the invention, the electromagnetic radiation is radar.
In another aspect of the invention, the sensor uses one of particle radiation, acoustic radiation, and magnetic resonance imaging.
In another aspect of the invention, the transfer unit includes one of a network, magnetic media, and optical media.
In another aspect of the invention, the display is one of a CRT, a printer, a volumetric display, a stereoscopic, and an immersion display.
These and other objects and embodiments of the invention will become readily apparent to those of ordinary skill in the art from the following description of the invention, which is to be read in conjunction with the accompanying drawings and appended claims.